harmonic coordinate finite element method for …trip.rice.edu/reports/2011/xin.pdfmotivation for...
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Harmonic Coordinate Finite Element Method forAcoustic Waves
Xin Wang
The Rice Inversion Project
Mar. 30th, 2012
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Outline
Introduction
Harmonic Coordinates FEM
Mass Lumping
Numerical Results
Conclusion
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Motivation
Scalar acoustic wave equation
1
κ
∂2u
∂t2−∇ · 1
ρ∇u = f
with appropriate boundary, initial conditions
Typical setting in seismic applications:
I heterogeneous κ, ρ with low contrast O(1)
I model data κ, ρ defined on regular Cartesian grids
I large scale ⇒ waves propagate O(102) wavelengths; solutionsfor many different f
I f smooth in time
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MotivationFor piecewise constant κ, ρ with interfaces
I FDM: first order interface error, time shift, incorrect arrivaltime, no obvious way to fix (Brown 84, Symes & Vdovina 09)
I Accuracy of standard FEM (eg specFEM3D) relies onadaptive, interface fitting meshes
I Exception: FDM derived from mass-lumped FEM on regulargrid for constant density acoustics has 2nd orderconvergence even with interfaces (Symes & Terentyev 2009)
I Aim of this project: modify FD/FEM with full accuracy forvariable density acoustics
0
1
2
depth
(km)
0 2 4 6 8offset (km)
1.5 2.0 2.5 3.0 3.5 4.0km/s
Figure: Velocity model4
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MotivationFor highly oscillating coefficient (rough medium)
e.g., coefficient varies on scale 1 m ⇒ accurate regular FDsimulations of 30 Hz waves may require 1 m grid though thecorresponding wavelength is about 100 m at velocity of 3 km/s
Can we create an accurate FEM on coarse ( wavelength) grid?0.51.01.52.02.5
Depth (km)
23
45
Velocity (km/sec)
A log of compressional wave velocity from a well in West Texas,supplied to TRIP by Total E&P USA and used by permission.
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Motivation
Ideal numerical method:
I high order
I no special meshing, i.e., on regular grids
I works for problems with heterogeneous media
Owhadi and Zhang 2007: 2D harmonic coordinate finite elementmethod, sub-optimal convergence on regular grids
Binford 11: showed using the true support recovers the optimalorder of convergence on triangular meshes for 2D static interfaceproblems, and mesh of diameter h2 is used to approximate theharmonic coordinates
Result of project: modification of Owhadi and Zhang’s methodwith full 2nd order convergence for variable density acoustic WE
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Outline
Introduction
Harmonic Coordinates FEM
Mass Lumping
Numerical Results
Conclusion
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Harmonic CoordinatesGlobal C -harmonic coordinates F in 2D, its componentsF1(x1, x2),F2(x1, x2) are weak sols of
∇ · C (x)∇Fi = 0 in Ω
Fi = xi on ∂Ω
F : Ω→ Ω C -harmonic coordinates
e.g.,
x2
x1C1 = 20
C2 = 1
r0 =1√
2π
(1, 1)
(−1,−1)
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Harmonic Coordinates
I physical regular grid (x1, x2) = (jhx , khy ) (left),
I harmonic grid (F1,F2) =(F1(jhx , khy ),F2(jhx , khy )
)(right)
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Harmonic Coordinate FEM
Workflow of HCFEM:
1 prepare a regular mesh on physical domain, T H ;
2 approximate F on a fine mesh T h by Fh
3 construct the harmonic triangulation T H = Fh(T H);
4 construct the HCFE spaceSH = spanφHi Fh : i = 0, · · · ,Nh, whereSH = spanφHi : i = 0, · · · ,Nh is Q1 FEM space on
harmonic grid TH ;
5 solve the original problem by Galerkin method on SH .
⇒ solve n (≤ 3) harmonic problems to obtain harmoniccoordinates F; resulting stiffness and mass matrices of HCFEMhave the same sparsity pattern as in standard FEM
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Harmonic Coordinate FEM
Galerkin approximation uh ∈ Sh (harmonic coordinate FE space)to u (weak solution of −∇ · C∇u = f ) has optimal error:∫ ∫
Ω|∇u −∇uh|2 dx = O(h2)
Why this is so: see WWS talk in 2008 TRIP review meeting
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Harmonic Coordinate FEM
For wave equation:
1
κ
∂2u
∂t2−∇ · 1
ρ∇u = f in Ω ⊂ R2
Dirichlet BC, u ≡ 0, t < 0, f ∈ L2([0,T ], L2(Ω))
Assume f is finite bandwidth ⇒ Galerkin approximation uh inHCFE space Sh on mesh of diam h has optimal orderapproximation in energy
e[u − uh](t) = O(h2), 0 ≤ t ≤ T
with e[u](t) :=1
2
(∫∫Ω dx | 1√
κ
∂u
∂t|2 + | 1
√ρ∇u|2
)(t)
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1D Illustration1D elliptic interface problem
(βux)x = f 0 ≤ x ≤ 1, u(0) = u(1) = 0
f ∈ L2(0, 1), β has discontinuity at x = α
β(x) =
β− x < αβ+ x > α
displacement u is continuous as well as normal stress βux at α
1D ’linear’ HCFE basis:
0.6 0.65 0.70
0.2
0.4
0.6
0.8
1
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Outline
Introduction
Harmonic Coordinates FEM
Mass Lumping
Numerical Results
Conclusion
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FEM Discretization for Acoustic WavesFor acoustic wave equation:
1
κ
∂2u
∂t2−∇ · 1
ρ∇u = f
FE space S = spanψj(x)Nhj=0, FE solution uh =
Nh∑j=0
uj(t)ψj(x).
FEM semi-discretization:
Mh d2Uh
dt2+ NhUh = F h
Uh(t) = [u0, ...uNh]T , F h
i =
∫Ωf ψi dx , Mh
ij =
∫Ω
1
κψiψj dx ,
Nhij =
∫Ω
1
ρ∇ψi · ∇ψj dx
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Mass Lumping
2nd order time discretization:
MhUh(t + ∆t)− 2Uh(t) + Uh(t −∆t)
∆t2+ NhUh(t) = F h(t)
⇒ every time update involves solving a linear system MhUh =RHS
Replace Mh by a diagonal matrix Mh,
Mhii =
∑j
Mhij
Can achieve optimal rate of convergence if the solution u issmooth (e.g., H2(Ω))
My thesis has the details for validation of mass lumping
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Outline
Introduction
Harmonic Coordinates FEM
Mass Lumping
Numerical Results
Conclusion
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Implementation and Computation
Implementation: based on deal.II, a C++ program library targetedat the computational solution of partial differential equations usingadaptive finite elements
I built-in quadrilateral mesh generation and mesh adaptivity
I various finite element spaces, DG
I interfaces for parallel linear system solvers (eg PETSc)
I data output format for quick view (eg paraview, opendx,gnuplot, ps)
Computation: using DAVinCI@RICE cluster, 2304 processor coresin 192 Westmere nodes (12 processor cores per node) at 2.83 GHzwith 48 GB of RAM per node (4 GB per core).
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Elliptic BVP - Square Circle Model
−∇ · C(x)∇u = −9r in Ω
where r =√
x2 + y 2
For piecewise const C(x) shown in the figure below, analytical solution:
u =1
C(x)(r 3 − r 3
0 )
x2
x1C1
C2
r0 =1√
2π
(1, 1)
(−1,−1)19
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High Contrast: C1 = 20,C2 = 1
10−3
10−2
10−1
100
10−3
10−2
10−1
100
101
grid size H
se
mi−
H1 e
rro
r
HCFEM
O(H)
10−3
10−2
10−1
100
10−6
10−5
10−4
10−3
10−2
10−1
100
grid size H
L2 e
rro
r
HCFEM
O(H2)
I HCFEM is applied on the physical grid of diameter H
I Harmonic coordinates are approximated on the locally refined grid, in which thegrid size is O(h) (h = H2) near interfaces.
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High Contrast: C1 = 20,C2 = 1
10−3
10−2
10−1
100
10−3
10−2
10−1
100
101
grid size H
se
mi−
H1 e
rro
r
Q1 FEM
O(H)
10−3
10−2
10−1
100
10−6
10−5
10−4
10−3
10−2
10−1
100
grid size H
L2 err
or
Q1 FEM
O(H2)
I Standard FEM is applied on the physical grid of diameter H
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Low Contrast: C1 = 2,C2 = 1
10−3
10−2
10−1
100
10−3
10−2
10−1
100
101
grid size H
sem
i−H
1 err
or
HCFEMO(H)
10−3
10−2
10−1
100
10−6
10−4
10−2
100
grid size H
L2 err
or
HCFEM
O(H2)
I HCFEM is applied on the physical grid of diameter H
I Harmonic coordinates are approximated on the locally refined grid, in which thegrid size is O(h) (h = H2) near interfaces.
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Low Contrast: C1 = 2,C2 = 1
10−3
10−2
10−1
100
10−3
10−2
10−1
100
101
grid size H
sem
i−H
1 err
or
Q1 FEM
O(H)
10−3
10−2
10−1
100
10−6
10−4
10−2
100
grid size H
L2 err
or
Q1 FEM
O(H2)
I Standard FEM is applied on the physical grid of diameter H
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2D Acoustic Wave Tests
Acoustic wave equation:
κ−1∂2u
∂t2−∇
(1
ρ∇u)
= 0
u(x , 0) = g(x , 0), ut(x , 0) = gt(x , 0)
with g(x , t) =1
rf
(t − r
cs
)and
f (t) =(
1− 2 (πf0 (t + t0))2)e−(πf0(t+t0))2
, f0 central frequency,
cs =
√κ(xs)
ρ(xs), t0 =
1.45
f0
The following examples similar to those in Symes and Terentyev,SEG Expanded Abstracts 2009
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Dip Model
Central frequency f0 = 10 Hz, xs = [−300√
3 m,−300 m]
x2
x1
[ρ1, c1] = [3000 kg/m3, 1.5 m/s]
[ρ2, c2] = [1500 kg/m3, 3 m/s]
(2 km,2 km)
(-2 km,-2 km)
xs
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Dip ModelQ1 FEM solution, regular grid quadrature (= FDM) - this isequivalent to using ONLY the node values on the regular grid todefine mass, stiffness matrices
Figure: T = 0.75 s
Entire domain
h e p
7.8 m 3.62e-1 -
3.9 m 9.30e-2 1.96
1.9 m 2.31e-2 2.01
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Dip ModelQ1 FEM solution, accurate quadrature for mass and stiffnessmatrices’ computation,
Q1 FEM with accurate quadrature is also (2,2) FDM but withdifferent coefficients - this is Igor’s result (constant densityacoustics).
Figure: T = 0.75 s
Entire domain
h e p
7.8 m 3.56e-1 -
3.9 m 9.13e-2 1.96
1.9 m 2.27e-2 2.01
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Dip Model
HCFEM solution
Figure: T = 0.75 s
Entire domain
h e p
7.8 m 3.63-1 -
3.9 m 9.36e-2 1.96
1.9 m 2.34e-2 2.01
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Dome Modelcentral frequency f0 = 15 Hz, xs = [3920 m, 3010 m]
0
2
4
6
8
0 2 4 6 8
1.5 2.0 2.5 3.0 3.5 4.0
Figure: Velocity model29
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Dome ModelDifference between HCFEM solution on regular grid (h = 7.8125m) and FEM solution on locally refined grid, same time stepping
Figure: T = 1.3 s
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Dome ModelDifference between FEM solution on regular grid (h = 7.8125 m)and FEM solution on locally refined grid, same time stepping
Figure: T = 1.3 s
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Outline
Introduction
Harmonic Coordinates FEM
Mass Lumping
Numerical Results
Conclusion
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Conclusion
2D harmonic coordinate finite element method (HCFEM) onregular grids achieves second order convergence rate for static anddynamic acoustic interface problems.
For dip model: HCFEM and mass lumping is at least as good asQ1 FEM with accurate quadrature, when density contrasts are low(typical of seismic). Both seem to get rid of stairstep diffractions(more or less). More refined analysis shows HCFEM somewhatmore accurate.
For dome model: HCFEM closer to refined-grid FEM when same(very short) time steps taken
Future work:
I Fill in the theoretical gaps,
I Extensions, eg higher order, DG, elasticity.
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